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Introduction to the Section Formula
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Today we're going to learn about the Section Formula, which helps us find the coordinates of a point that divides a line segment in a specific ratio. To illustrate this, consider two points, A and B, on a coordinate plane.
So, how exactly does the ratio affect where the point will be located?
Great question! If P divides AB in the ratio of m:n, the distance from A to P will be m parts, and from P to B will be n parts. Hence, the coordinates of P are calculated using the formula.
Can you remind us what the formula looks like?
Certainly! The formula is: P(x, y) = \left( \frac{mx_{2} + nx_{1}}{m + n}, \frac{my_{2} + ny_{1}}{m + n} \right). Remember, xβ and yβ are the coordinates for A, while xβ and yβ are for B. An easy way to remember the formula is by the acronym 'MINE' for mxβ + nxβ!
I like that! MINE helps me remember which coordinates go where.
Exactly! Let's summarize: the Section Formula helps calculate positions based on ratios by using coordinates from two endpoints. In the following sessions, we'll apply this formula with examples.
Derivation of the Section Formula
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Now, let's derive the Section Formula. We use similar triangles to develop it. If you draw perpendicular lines from points A and B to the axes, two right triangles are created. Do you recall the AA similarity criterion?
Yes! Two triangles are similar if they share an angle and have the other angles equal.
Well done! We can set up ratios of corresponding sides which gives us the relationships needed for our formula. Specifically, the segments related through y-coordinates will equal the segments related through the x-coordinates based on the ratio.
So that means we set up equations using those ratios to find x and y coordinates, right?
Exactly! Once we establish those ratios, we end up with a systematic way to find any point P that divides AB. Let's wrap this session with the reminder that the ratios dictate the relative distances of segments.
Example Problem
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Let's work through an example: Find the coordinates of the point P which divides the segment between (4, -3) and (8, 5) in the ratio 3:1.
So, what do we plug into the formula?
Exactly! Here, A(4, -3) will be (xβ, yβ) and B(8, 5) will be (xβ, yβ). The m:n ratio means m=3 and n=1. Plugging them into the formula gives us:
I'll calculate that! So, x = \frac{3(8) + 1(4)}{3 + 1} = \frac{24 + 4}{4} = 7, and for y, it's y = \frac{3(5) + 1(-3)}{3 + 1} = \frac{15 - 3}{4} = 3. Right?
Spot on! The coordinates of P are (7, 3). This clearly demonstrates how applying the Section Formula yields precise results. Summarizing, we can always use ratios and the formula to identify any dividing point between segments.
Applications of the Section Formula
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Alright, class! Now letβs discuss where we can apply the Section Formula in real-world situations.
Does it relate to anything specific, like in architecture or engineering?
Absolutely! In architecture, it helps in placing structural supports proportionately. Also in computer graphics, it is used to divide lines and shapes correctly.
I can see that being really useful! Can we use it for any ratios, even non-integer ones?
Yes indeed! You can use any positive ratio, whether it's a whole number, fraction, or decimal. This flexibility allows for diverse applications in various fields. To recap, Section Formula is not just a mathematical tool but a real-world application too!
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
This section introduces the Section Formula, which enables the calculation of the coordinates of a point that divides a line segment defined by two endpoints. The ratio in which the segment is divided can be specified, allowing for both internal and external division calculations. The section also discusses special cases and provides examples to illustrate the application of the formula.
Detailed
Section Formula
In this section, we explore the Section Formula which allows us to find the coordinates of a point that divides a line segment into a specified ratio. Given two points, A(xβ, yβ) and B(xβ, yβ), the point P(x, y) that divides the segment AB in the ratio m:n can be calculated using the formulas:
$$
P(x, y) = \left( \frac{mx_{2} + nx_{1}}{m + n}, \frac{my_{2} + ny_{1}}{m + n} \right)
$$
Key Points:
- Understanding Ratios: Ratios dictate how two segments relate to each other β if P divides AB in the ratio 1:2, this means the section closer to A is one part while the section closer to B is two parts.
- Similar Triangles: The derivation of the formula is based on the properties of similar triangles, which maintain the proportional relationship regardless of triangle orientation.
- Special Cases: The midpoint of a segment is a special case of the section formula where the ratio is 1:1.
- Internal vs. External Division: The section formula primarily deals with internal divisions, where P lies within the segment AB, but can also apply to external divisions where P lies outside between A and B when extended.
Overall, understanding the Section Formula is crucial for solving various geometric problems efficiently and forms a foundational concept in coordinate geometry.
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Introduction to the Section Formula
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Let us recall the situation in Section 7.2. Suppose a telephone company wants to position a relay tower at P between A and B is such a way that the distance of the tower from B is twice its distance from A. If P lies on AB, it will divide AB in the ratio 1 : 2 (see Fig. 7.9). If we take A as the origin O, and 1 km as one unit on both the axis, the coordinates of B will be (36, 15). In order to know the position of the tower, we must know the coordinates of P. How do we find these coordinates?
Detailed Explanation
In this part of the section, we set up a scenario where a relay tower needs to be placed between two points A and B. Point P divides the distance between A and B such that the distance from B to P is twice as long as A to P. This means we can represent this condition as a ratio of distances. For practical calculations, A is taken as the origin point (0,0), and the coordinates for B are given as (36, 15). Understanding how P divides the segment allows us to calculate its coordinates.
Examples & Analogies
Think of this like placing a light post along a street that connects two houses. If you want the light post to be closer to house A than to house B, you can think of it needing to be placed 'two-thirds of the way' from A towards B.
Finding the Coordinates of Point P
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Let the coordinates of P be (x, y). Draw perpendiculars from P and B to the x-axis, meeting it in D and E, respectively. Draw PC perpendicular to BE. Then, by the AA similarity criterion, studied in Chapter 6, ΞPOD and ΞBPC are similar. Therefore, OD / OP = 1 / 2 and PD / OP = 1 / 2.
Detailed Explanation
Here, we denote the coordinates of point P as (x,y). To find the coordinates mathematically, we use similar triangles. By drawing perpendiculars, we can set up relationships between the segments created by point P falling on the line joining A and B, thus enabling us to create equations that represent the proportionate distances dictated by the ratios stated earlier.
Examples & Analogies
Imagine you have a right triangle formed by an upright pole leading down to the ground. You can measure the height at which the pole intersects with the ground to know how high up the pole is placed, just as we determine the position of point P based on the conditions set in the question.
General Formula for Internal Division
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Now let us use the understanding that you may have developed through this example to obtain the general formula. Consider any two points A(x1, y1) and B(x2, y2) and assume that P (x, y) divides AB internally in the ratio m : n, i.e., PA : PB = m : n. The coordinates of the point P(x, y) which divides the line segment joining the points A(x1, y1) and B(x2, y2) internally in the ratio m : n are given by:
\[ P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
Detailed Explanation
The formula developed here allows us to find the coordinates of any point P that divides the line segment connecting points A and B in any given ratio, m:n. It combines the coordinates of both points, weighted by the ratio in which P divides the segment, providing a powerful tool for analyzing geometric relationships in coordinate geometry.
Examples & Analogies
It's like if you wanted to find a specific point along a path where two friends are walking. If one friend is farther away than the other based on their distances apart, you could calculate that middle point using their relative positions, ensuring that you meet in the exact place you agreed upon!
Special Case: Mid-point of a Line Segment
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Special Case: The mid-point of a line segment divides the line segment in the ratio 1 : 1. Therefore, the coordinates of the mid-point P of the join of the points A(x1, y1) and B(x2, y2) is
\[ P(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Detailed Explanation
This special case highlights how to find the midpoint of a line segment which is simply the average of the respective coordinates of the endpoints A and B. This is a specific application of the section formula where both parts of the division are equal, leading to straightforward calculations.
Examples & Analogies
Consider a scenario where you're meeting a friend halfway across a park. Finding that halfway point involves literally just averaging the two locationsβit's the easiest way to ensure you both arrive at the same time!
Examples of Using the Section Formula
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Let us solve a few examples based on the section formula. Example 6: Find the coordinates of the point which divides the line segment joining the points (4, β3) and (8, 5) in the ratio 3 : 1 internally. Solution: Let P(x, y) be the required point. Using the section formula, we get:
\[ x = \frac{3(8) + 1(4)}{3 + 1}, y = \frac{3(5) + 1(-3)}{3 + 1} \]
Thus, P = (7, 3).
Detailed Explanation
In this section, we go through practical examples that show how the section formula is applied. By plugging values into the formula derived earlier, we can calculate exact points along the line segment connecting two given coordinates, refreshing the importance of the formula in applied geometry.
Examples & Analogies
Visualize a bus route with stops A and B. If you want to know the location of a new stop that should be three times closer to A than to B, you could use this formula to find out where exactly to place it, ensuring the efficiency of the bus service.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Ratio: Indicates the proportional relationship between two segments.
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Coordinates: The x and y values that determine the position of a point on a grid.
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Internal Division: Refers to dividing a segment where the dividing point lies between the two endpoints.
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External Division: Refers to dividing a segment where the dividing point lies beyond the endpoints.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example: For points (4, -3) and (8, 5) dividing in the ratio 3:1, the coordinates would be calculated using the Section Formula to give (7, 3).
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Example: To find a point on the line segment between (2, -3) and (4, 2) that divides it into equal parts, you'll calculate the midpoint using the section formula with ratio 1:1.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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To find a point on the line, use the Section Formula, make it fine!
π Fascinating Stories
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A builder had two points A and B. If he wanted to place a beam at an exact spot, he used the Section Formula to find the perfect point, making his construction straight and true.
π§ Other Memory Gems
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MINE for mxβ + nxβ indicates the sections' mix!
π― Super Acronyms
PRACTICE
- Position Ratio And Coordinates To Internal Coordinate Extract!
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Section Formula
Definition:
A formula used to find the coordinates of a point that divides a line segment into a specified ratio.
-
Term: Ratio
Definition:
A relationship between two numbers indicating how many times the first number contains the second.
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Term: Similar Triangles
Definition:
Triangles that have the same shape but may have different sizes; their corresponding angles are equal.
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Term: Internal Division
Definition:
When a point divides a line segment such that it falls between the two endpoints.
-
Term: External Division
Definition:
When a point divides a line segment such that it lies outside the two endpoints.
Similar Question
Example 11 : If the points E(2, 3), F(4, 5), G(7, 1), and H(p, 2) are the vertices of a parallelogram, taken in order, find the value of $p$.
Solution : We know that diagonals of a parallelogram bisect each other.
So, the coordinates of the mid-point of EF = coordinates of the mid-point of GH.
Thus:
\[ \left( \frac{2 + 4}{2}, \frac{3 + 5}{2} \right) = \left( \frac{7 + p}{2}, \frac{1 + 2}{2} \right) \]
This gives us:
\[ \left( 3, 4 \right) = \left( \frac{7 + p}{2}, \frac{3}{2} \right) \]
From the $y$-coordinate, we have:
\[ 4 = \frac{3}{2} \]
By solving the $x$-coordinate:
\[ 3 = \frac{7 + p}{2} \implies 6 = 7 + p \implies p = -1 \]
So, the value of $p$ is -1.